L(s) = 1 | + (−0.101 − 0.0738i)3-s + (−0.809 + 0.587i)4-s + (−0.263 + 0.809i)5-s + (1.60 − 1.16i)7-s + (−0.304 − 0.936i)9-s + (0.535 + 0.844i)11-s + 0.125·12-s + (−0.393 − 1.21i)13-s + (0.0865 − 0.0628i)15-s + (0.309 − 0.951i)16-s + (−0.263 − 0.809i)20-s − 0.249·21-s + (0.222 + 0.161i)25-s + (−0.0770 + 0.236i)27-s + (−0.613 + 1.88i)28-s + ⋯ |
L(s) = 1 | + (−0.101 − 0.0738i)3-s + (−0.809 + 0.587i)4-s + (−0.263 + 0.809i)5-s + (1.60 − 1.16i)7-s + (−0.304 − 0.936i)9-s + (0.535 + 0.844i)11-s + 0.125·12-s + (−0.393 − 1.21i)13-s + (0.0865 − 0.0628i)15-s + (0.309 − 0.951i)16-s + (−0.263 − 0.809i)20-s − 0.249·21-s + (0.222 + 0.161i)25-s + (−0.0770 + 0.236i)27-s + (−0.613 + 1.88i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9915454837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9915454837\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.535 - 0.844i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 1.75T + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.704708442329950532809938594200, −8.865677225398879385161814528654, −7.72776061496307755923344039165, −7.65884067175106104011255754416, −6.68502879272659510890442279987, −5.38791945371876433586444349833, −4.46564662686727655071980929026, −3.86596972146155203400551304754, −2.82338287966799186893667330868, −1.06009824413208863222803088997,
1.33723571415909356380006815584, 2.35352747654180101931699450028, 4.26499465287914367144632120587, 4.69780851911102491214122188223, 5.45188911792079692250344237158, 6.07818434877950617814977123566, 7.68393675045939453450336042442, 8.310329069737734145397489889785, 9.103936171883904916611135227760, 9.190811396558022731939734285519